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\title{实变函数第四章：可测函数 }
\author{CQX ET AL}

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\begin{frame}{第四章目录 }

\begin{enumerate}

\item[4.1.] 可测函数及其性质
\item[4.2.] 叶戈罗夫定理
\item[4.3.] 可测函数的构造
\item[4.4.] 依测度收敛


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%\begin{frame}{第四章重点 }
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\begin{frame}{4.1.1. 可测函数及其性质  }

\begin{itemize}

\item  {\color{red}问题：实变函数研究的函数的定义域和值域是什么？ }

\item  解答：实变函数研究的函数的定义域是可测集 $E\subseteq\mathbb{R}^n$, 值域是 $\mathbb{R}\cup \{\pm\infty\}$. 


\end{itemize}

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\begin{frame}{4.1.2.  }

\begin{itemize}

\item  {\color{red}问题：设 $f$ 是定义在可测集 $E\subseteq\mathbb{R}^n$ 的实函数？什么时候称 $f$ 是可测函数？ }

%\item  解答：


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\begin{frame}{4.1.3. 定理1 }

\begin{itemize}

\item  {\color{red}问题：设 $f$ 是定义在可测集 $E\subseteq\mathbb{R}^n$ 的实函数。证明下述条件相互等价： }
\begin{enumerate}
\item  {\color{red}$f$ 是可测函数。 }
\item  {\color{red}对任意实数 $a$, $E[f\ge a]$ 都是可测集。 }
\item  {\color{red}对任意实数 $a$, $E[f<a]$ 都是可测集。 }
\item  {\color{red}对任意实数 $a$, $E[f\le a]$ 都是可测集。 }
\item  {\color{red}对任意实数 $a,b, (a<b)$, $E[a\le f<b]$ 都是可测集。 }
\end{enumerate}

%\item  解答：


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\begin{frame}{4.1.4.  }

\begin{itemize}

\item  {\color{red}问题：设 $f(x)$ 是 $E$ 上的可测函数，设 $a$ 是任意实数。证明 $E[f=a]$ 是可测集。 }

%\item  解答：


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\begin{frame}{4.1.5.  }

\begin{itemize}

\item  {\color{red}问题：证明区间 $[a,b]$ 上的连续函数和单调函数都是可测函数。  }

%\item  解答：


\end{itemize}

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\begin{frame}{4.1.6.  }

\begin{itemize}

\item  {\color{red}问题：设 $E\subseteq\mathbb{R}^n$ 是可测集。设 $f:E\to\mathbb{R}\cup\{\pm\infty\}$ 是实函数。什么时候称 $f$ 是一个连续函数？ }

%\item  解答：


\end{itemize}

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\begin{frame}{4.1.7. 定理2 }

\begin{itemize}

\item  {\color{red}问题：证明可测集 $E\subseteq\mathbb{R}^n$ 上的连续函数是可测函数。 }

%\item  解答：


\end{itemize}

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\begin{frame}{4.1.8. 定理3 }

\begin{itemize}

\item  {\color{red}问题：证明： }
\begin{enumerate}
\item  {\color{red}设 $f(x)$ 是可测集 $E$ 上的可测函数，设 $E_1\subseteq E$ 是可测子集，则 $f(x)$ 看作是定义在 $E_1$ 上的函数时，也是可测函数。  }
\item  {\color{red}设 $f(x)$ 定义在有限个可测集 $E_i (1\le i\le s)$ 的并集 $E$ 上，并且 $f(x)$ 看作定义在每个 $E_i$ 上的函数时都是可测的，则 $f(x)$ 看作定义在 $E$ 上的函数时也是可测的。  }
\end{enumerate}


%\item  解答：


\end{itemize}

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\begin{frame}{4.1.9.  }

\begin{itemize}

\item  {\color{red}问题：设 $f$ 是定义在可测集 $E\subseteq\mathbb{R}^n$ 的实函数。什么时候称 $f(x)$ 是简单函数？ }

%\item  解答：


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\begin{frame}{4.1.10.  }

\begin{itemize}

\item  {\color{red}问题：设 $f(x)$ 与 $g(x)$ 都是 $E$ 上的可测函数。证明 $E[f>g]$ 与 $E[f\ge g]$ 都是可测集。 }

%\item  解答：


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\begin{frame}{4.1.11. 定理4 }

\begin{itemize}

\item  {\color{red}问题：设 $f(x),g(x)$ 是可测集 $E$ 上的可测函数。则 $$f(x)+g(x),\,\, |f(x)|,\,\, \frac{1}{f(x)},\,\, f(x)g(x)$$ 也都是 $E$ 上的可测函数。 }

%\item  解答：


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\begin{frame}{4.1.12. 定理5 }

\begin{itemize}

\item  {\color{red}问题：设 $\{f_n(x)\}$ 是可测集 $E$ 上的一列可测函数。则 
$$\mu(x)=\inf f_n(x) \,\,\text{ 与 }\,\, \lambda(x) = \sup f_n(x)$$ 都是 $E$ 上的可测函数。 }

%\item  解答：


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\begin{frame}{4.1.13. 定理6 }

\begin{itemize}

\item  {\color{red}问题：设 $\{f_n(x)\}$ 是可测集 $E$ 上的一列可测函数。则 
$$ F(x)=\varliminf_{n\to\infty} f_n(x) \,\,\text{ 与 }\,\, G(x) = \varlimsup_{n\to\infty} f_n(x)$$ 都是 $E$ 上的可测函数。 }

%\item  解答：


\end{itemize}

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\begin{frame}{4.1.14. 定理7 }

\begin{itemize}

\item  {\color{red}问题：证明： }
\begin{enumerate}
\item  {\color{red}设 $f(x)$ 是可测集 $E$ 上的非负可测函数，则存在可测的简单函数序列 $\varphi_k(x)$, 使得对任意 $x\in E$, $\varphi_k(x)\le \varphi_{k+1}(x), (k=1,2,\cdots)$, 且 $\lim\limits_{k\to\infty} \varphi_k(x)=f(x).$    }
\item  {\color{red} 设 $f(x)$ 是可测集 $E$ 上的可测函数，则存在可测的简单函数序列 $\varphi_k(x)$, 使得对任意 $x\in E$, $\varphi_k(x)\le \varphi_{k+1}(x), (k=1,2,\cdots)$, 且 $\lim\limits_{k\to\infty} \varphi_k(x)=f(x).$ 
若 $f(x)$ 还在 $E$ 上有界，则上述收敛可以是一致的。}
\end{enumerate}

%\item  解答：


\end{itemize}

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\begin{frame}{4.1.15.  }

\begin{itemize}

\item  {\color{red}问题：称一个与可测集 $E$ 有关的命题在 $E$ 上几乎处处成立，指的是什么？ }

%\item  解答：


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\begin{frame}{4.1.16.  }

\begin{itemize}

\item  {\color{red}问题：判断下述命题是否正确： }
\begin{enumerate}
\item  {\color{red}  $|\tan(x)|<\infty$ 在 $E\mathbb{R}$ 上几乎处处成立。  }
\item  {\color{red}  定义在区间 $E=[0,1]$ 上的狄利克雷函数 $D(x)$ 几乎处处等于零。  }
\end{enumerate}



%\item  解答：


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\begin{frame}{4.1.17.  }

\begin{itemize}

\item  {\color{red}问题：设 $f(x),g(x),h(x)$ 都是定义在 $E$ 上的实函数。若 $f(x)$ 与 $g(x)$ 在 $E$ 上几乎处处相等， $g(x)$ 与 $h(x)$ 在 $E$ 上几乎处处相等，则 $f(x)$ 与 $h(x)$ 在 $E$ 上也几乎处处相等。 }

%\item  解答：


\end{itemize}

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\begin{frame}{4.2.1. 叶戈罗夫定理  }

\begin{itemize}

\item  {\color{red}问题：设 $E$ 是有限测度的可测集。设 $f_n(x)$ 是 $E$ 上的一列可测函数，且在 $E$ 上几乎处处收敛于有限函数 $f(x)$. 则对任意 $\delta>0$, 存在子集 $E_\delta\subseteq E$, 使得 $m(E-E_\delta)<\delta$, 且 $\{f_n(x)\}$ 在 $E_\delta$ 上是一致收敛的。 }

%\item  解答：一个例子是 $E=[0,1], \,\, f_n(x) = x^n$. 


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\begin{frame}{4.3.1. 可测函数的构造  }

\begin{itemize}

\item  {\color{red}卢津定理1：设 $f(x)$ 是 $E$ 上几乎处处有限的可测函数，则对任意 $\delta>0$, 存在闭子集 $F_\delta\subseteq E$, 使得 $f(x)$ 在 $F_\delta$ 上是连续函数，且 $m(E-F_\delta)<\delta$.  }

%\item  解答：


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\begin{frame}{4.3.2.  }

\begin{itemize}

\item  {\color{red}卢津定理2：设 $f(x)$ 是 $E\subseteq \mathbb{R}$ 上几乎处处有限的可测函数，则对任意 $\delta>0$, 存在闭子集 $F\subseteq E$ 以及整个 $\mathbb{R}$ 上的连续函数 $g(x)$, 使得在 $F$ 上 $g(x)=f(x)$, 且 $m(E-F)<\delta$. 
此外还可要求 $\sup_\mathbb{R}\{g(x)\} = \sup_\mathbb{R}\{f(x)\}$ 以及 $\inf\mathbb{R}\{g(x)\} = \inf\mathbb{R}\{f(x)\}$. }

%\item  解答：


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\begin{frame}{4.4.1. 依测度收敛  }

\begin{itemize}

\item  {\color{red}问题：设 $\{f_n\}$ 是可测集 $E$ 上的一列几乎处处有限的可测函数。
设 $f(x)$ 是 $E$ 上几乎处处有限的可测函数。
什么时候称 $\{f_n(x)\}$ 依测度收敛于 $f(x)$ ？ }

%\item  解答：


\end{itemize}

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\begin{frame}{4.4.2.  }

\begin{itemize}

\item  {\color{red}问题：举例说明函数列可以处处不收敛，但是是依测度收敛的。 }

%\item  解答：


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\begin{frame}{4.4.3. 定理1 }

\begin{itemize}

\item  {\color{red}里斯定理：设可测集 $E$ 上的函数列 $f_n(x)$ 依测度收敛于 $f(x)$, 则存在子列 $f_{n_i}(x)$ 在 $E$ 上几乎处处收敛于 $f(x)$. }

%\item  解答：


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\begin{frame}{4.4.4. 定理2 }

\begin{itemize}

\item  {\color{red}勒贝格定理：设 $E$ 是有限测度的可测集，$\{f_n(x)\}$ 是 $E$ 上的几乎处处有限的可测函数列，$f(x)$ 是 $E$ 上的几乎处处有限的函数，且 $\{f_n(x)\}$ 在 $E$ 上几乎处处收敛于 $f(x)$. 则 $f_n(x)$ 依测度收敛于 $f(x)$.   }

%\item  解答：


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\begin{frame}{4.4.5. 定理3 }

\begin{itemize}

\item  {\color{red}问题：设 $\{f_n(x)\}$ 在 $E$ 上依测度收敛于 $f(x)$, 也依测度收敛于 $g(x)$, 则 $f(x)=g(x)$ 在 $E$ 上几乎处处成立。 }

%\item  解答：


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\begin{frame}{4.4.6.  }

\begin{itemize}

\item  {\color{red}问题：设 $E\subseteq\mathbb{R}$, 设 $f(x)$ 是 $E$ 上几乎处处有限的可测函数。证明：存在定义在 $\mathbb{R}$ 上的一列连续函数 $g_n(x)$, 使得 $\lim\limits_{n\to\infty} g_n(x)=f(x)$ 在 $E$ 上几乎处处成立。}

%\item  解答：


\end{itemize}

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\begin{frame}{习题1  }

\begin{itemize}

\item  {\color{red}问题：证明 $f(x)$ 在 $E$ 上可测的充分必要条件是对任意有理数 $r$, 集合 $E[f>r]$ 是可测集。
如果集合 $E[f=r]$ 可测，问 $f(x)$ 是否可测？ }

%\item  解答：


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\begin{frame}{习题3  }

\begin{itemize}

\item  {\color{red}问题：设 $E$ 是 $[0,1]$ 中的不可测集。设 $f(x)=\left\{\begin{array}{ll} x, & x\in E, \\ -x, & x\in [0,1]-E, \end{array}\right. $ 问 $f(x)$ 在 $[0,1]$ 上是否可测？问 $|f(x)|$ 是否可测？ }

%\item  解答：


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\begin{frame}{习题6  }

\begin{itemize}

\item  {\color{red}问题：设 $f(x)$ 在 $(-\infty,\infty)$ 上连续，设 $g(x)$ 在 可测集 $E\subseteq\mathbb{R}^n$ 上有限可测，则 $f(g(x))$ 在 $E$ 上可测。}

%\item  解答：


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\begin{frame}{习题9  }

\begin{itemize}

\item  {\color{red}问题：设在可测集 $E$ 上，$\{f_n(x)\}$ 依测度收敛于 $f(x)$, 且对任意正整数 $n$, $f_n(x)\le g(x)$ 在 $E$ 上几乎处处成立。证明 $f(x)\le g(x)$ 在 $E$ 上几乎处处成立。 }

%\item  解答：


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\begin{frame}{习题13  }

\begin{itemize}

\item  {\color{red}问题：设 $m(E)<\infty$. 设 $\{f_n(x)\}$ 与 $\{g_n(x)\}$ 是 $E$ 上的两个有限的可测函数列，分别依测度收敛于 $f(x)$ 与 $g(x)$. 证明 $\{f_n(x)g_n(x)\}$ 依测度收敛于 $f(x)g(x)$.  }

%\item  解答：


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\begin{frame}{习题16  }

\begin{itemize}

\item  {\color{red}问题：设 $m(E)<\infty$, 设 $\{f_n(x)\}$ 是 $E$ 上有限的可测函数列。
记 $$g_n(x) = \sup\{ |f_k(x)|: k\ge n\},$$
证明 $\lim\limits_{n\to\infty} f_n(x)=0$ 在 $E$ 上几乎处处成立的充分必要条件是 $\{g_n(x)\}$ 依测度收敛于零。
 }

%\item  解答：


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